5777 &
5993
are the only known odd numbers that are not prime and cannot be written as sum of a prime and twice a square (p + 2*n^2, n integer).
Prime numbers with latter property are 'Stern primes' 2, 3, 17, 137, 227, 977, 1187, 1493 (Moritz Abraham Stern, 1807–1894),
no more found, checked until ∼ 10^9 (M.F. Hasler 2007). It was one of Goldbach's conjectures.
more interesting 4-digit numbers:
(background-color: #005777)